| The goal of this research is to study the behavior of a group of vehicles that are controlled by virtual forces. The process involved includes the development of appropriate system equations and preliminary control structure, the analysis of the developed system, and analysis of subsequent changes in qualitative behavior of the system dynamics due to improvements of the control structure. In developing the system equations, two-dimensional kinematic models are used to describe each vehicle in the formation. Also, geometric relationships between the vehicles are established so that they can be used in the control of inter-vehicle spacing. Analysis is first conducted on a two-vehicle system traveling at constant velocity whose inputs are steering angles. The initial control scheme uses proportional and derivative feedback that keeps the vehicles at a specified separation distance. Solution perturbation method is used for stability analysis since the nonlinear system has a definable steady-state solution. Results indicate the existence of periodic solutions with a degeneracy occurring when the vehicles are aligned to follow one another. In these cases, an assumption included in the control scheme may be violated resulting in unstable behavior. Thus, the control scheme was revised to include a virtual force mapping to steering angle. When the analysis was extended to systems with a larger number of vehicles, solution perturbation was no longer a practical analysis method. Therefore, a numerical approach was used to calculate Lyapunov exponents since the simulation code had been verified through the two-vehicle analysis. The results of the Lyapunov exponent analysis proved that chaotic behavior occurs naturally as the number of vehicles in the system is increased. After coming to this conclusion, exploration of different virtual force functions was conducted to determine their effects on the system behavior. The first change was to add local neighbor coefficients to the virtual force function. This led to the stabilization of systems that were previously chaotic in nature. Next, a nonlinear damping function was used to induce chaotic behavior in systems where it did not occur naturally. Thus, through the proper selection of virtual force functions, chaotic behavior can be prevented or created. |
